The tree \(F(\Omega )\)  inherits from \(T_{1}^{\prime }\times T_{2}^{\prime }\)  an isometric action of \(\pi _{1}(S^{\prime })\) . We can define a length function
\(\ell : \pi _{1}(S^{\prime }) &\rightarrow \mathbb {R}^{+} \\\gamma &\mapsto \lim _{n \rightarrow +\infty }\frac{1}{n}d_{0}(x, \gamma ^{n}\cdot x)\) 

where \(x\)  is any point in \(F(\Omega )\)  (the definition is independent of the choice of \(x\) ). Since the action of \(\pi _{1}(S^{\prime })\)  on \(F(\Omega )\)  is minimal and irreducible, by [1]}, the isometry class of \((F(\Omega ), d_{0})\)  is completely determined by its length function. However, it is clear from the definition of \(\ell \)  and \(d_{0}\)  that \(\ell =\ell _{0}:=\ell _{1}+\ell _{2}\) , where \(\ell _{j}\)  denotes the analogously defined length functions on \(T_{1}^{\prime }\)  and \(T_{2}^{\prime }\) . On the other hand, \(\ell _{0}\)  is exactly the length function of the tree dual to the measured lamination \(\lambda _{0}\) , and the claim follows.
